Hacking the firmware seemed like a very complicated way to obtain a result, which could make this project very time-consuming, possibly grinding to a halt.
So I decided I wanted to give another go at the hard way, i.e. desoldering the three DIP chips that were controlling the LED driving circuitry and put my hardware in the middle.
Thanks to this Hackaday article and my hot air gun I successfully managed to desolder the MCU in its 40-pin DIP and two more logic chips.
Now it's just a matter of adding an Arduino to the right pins, to see if I can indeed control the row and column drivers.
If this is successful as well, I can replace the Arduino with a bunch of level shifters connected to the Linux embedded board.